where \(({\bf F})_{ij} = {\rm e}^{\psi _i\tilde{\delta }_j}\)  and \(\psi _i\) 
and \(\tilde{\delta }_j\)  are the eigenvalues of \(\Psi \)  and \(\widetilde{\Delta }\) , respectively.
In (REF ) we first used the HCIZ integral [1]}
and then the equality \(|{\cal U}(b)|= |{\cal S}(b,n)||{\cal U}(b-n)|\) .
Then, we apply twice the limit in (REF )
and obtain:
\(\lim _{{\bf q}\rightarrow 0}\lim _{{\bf p}\rightarrow {\bf 0}} \frac{|{\bf F}|}{{\cal V}(\Psi ){\cal V}(\widetilde{\Delta })} &=&\frac{\pi _b\Gamma _m(b)|{\bf Y}{\bf Y}\mathsf {^H}|^{m-b}}{\pi _m\Gamma _b(b){\cal V}({\bf Y}{\bf Y}\mathsf {^H})} \nonumber \\&&\quad \cdot \frac{\pi _b\Gamma _n(b)|\widehat{{\bf F}}||\Delta |^{n-b}}{\pi _n\Gamma _b(b){\cal V}(\Delta )} \) 
